A method of moments estimator of tail dependence

In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly multivariate setting. We consider a semi-parametric model in which the stable tail dependence function is parametrically modeled. Given a random sample from a bivariate distribution function, the problem is to estimate the unknown parameter. A method of moments estimator is proposed where a certain integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. Under very weak conditions, the estimator is shown to be consistent and asymptotically normal. Moreover, a comparison between the parametric and nonparametric estimators leads to a goodness-of-fit test for the semiparametric model. The performance of the estimator is illustrated for a discrete spectral measure that arises in a factor-type model and for which likelihood-based methods break down. A second example is that of a family of stable tail dependence functions of certain meta-elliptical distributions.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ130 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotic normality of extreme value estimators on C[0,1]C[0,1]

Consider $n$ i.i.d. random elements on $C[0,1]$. We show that, under an appropriate strengthening of the domain of attraction condition, natural estimators of the extreme-value index, which is now a continuous function, and the normalizing functions have a Gaussian process as limiting distribution. A key tool is the weak convergence of a weighted tail empirical process, which makes it possible to obtain the results uniformly on $[0,1]$. Detailed examples are also presented.Comment: Published at http://dx.doi.org/10.1214/009053605000000831 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Central limit theorems for local empirical processes near boundaries of sets

We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with $n$ and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ283 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Statistics of extremes under random censoring

We investigate the estimation of the extreme value index when the data are subject to random censorship. We prove, in a unified way, detailed asymptotic normality results for various estimators of the extreme value index and use these estimators as the main building block for estimators of extreme quantiles. We illustrate the quality of these methods by a small simulation study and apply the estimators to medical data.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ104 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Estimating the maximum possible earthquake magnitude using extreme value methodology: the Groningen case

The area-characteristic, maximum possible earthquake magnitude $T_M$ is required by the earthquake engineering community, disaster management agencies and the insurance industry. The Gutenberg-Richter law predicts that earthquake magnitudes $M$ follow a truncated exponential distribution. In the geophysical literature several estimation procedures were proposed, see for instance Kijko and Singh (Acta Geophys., 2011) and the references therein. Estimation of $T_M$ is of course an extreme value problem to which the classical methods for endpoint estimation could be applied. We argue that recent methods on truncated tails at high levels (Beirlant et al., Extremes, 2016; Electron. J. Stat., 2017) constitute a more appropriate setting for this estimation problem. We present upper confidence bounds to quantify uncertainty of the point estimates. We also compare methods from the extreme value and geophysical literature through simulations. Finally, the different methods are applied to the magnitude data for the earthquakes induced by gas extraction in the Groningen province of the Netherlands

Asymptotically distribution-free goodness-of-fit testing for tail copulas

Let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima $\bigvee_{i=1}^nX_i$ and $\bigvee_{i=1}^nY_i$ is then characterized by the marginal extreme value indices and the tail copula $R$. We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula $R$. The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric estimator of $R$. The transformed empirical process converges weakly to a standard Wiener process, paving the way for a multitude of asymptotically distribution-free goodness-of-fit tests. We also extend our results to the $m$-variate ($m>2$) case. In a simulation study we show that the limit theorems provide good approximations for finite samples and that tests based on the transformed empirical process have high power.Comment: Published at http://dx.doi.org/10.1214/14-AOS1304 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution

Consider a random sample from a bivariate distribution function $F$ in the max-domain of attraction of an extreme-value distribution function $G$. This $G$ is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of $F$. A major issue in multivariate extreme-value theory is the estimation of the spectral measure $\Phi_p$ with respect to the $L_p$ norm. For every $p\in[1,\infty]$, a nonparametric maximum empirical likelihood estimator is proposed for $\Phi_p$. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice.Comment: Published in at http://dx.doi.org/10.1214/08-AOS677 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

An M-estimator for tail dependence in arbitrary dimensions

Consider a random sample in the max-domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model. A new estimator for the unknown parameter is defined as the value that minimizes the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts. The minimization problem has, with probability tending to one, a unique, global solution. The estimator is consistent and asymptotically normal. The spectral measures of the tail dependence models to which the method applies can be discrete or continuous. Examples demonstrate the applicability and the performance of the method.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1023 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org